11-2 Areas of Trapezoids, Rhombi, and Kites
Find the area of each trapezoid, rhombus, or kite.
1. SOLUTION: ANSWER: 2. SOLUTION: ANSWER: 3. SOLUTION: ANSWER: eSolutions Manual - Powered by Cognero
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4. SHORT RESPONSE Suki is doing fashion design at 4-H Club. Her first project is to make a simple A-line skirt.
How much fabric will she need according to the design at the right?
ANSWER: 11-2 Areas of Trapezoids, Rhombi, and Kites
3. SOLUTION: ANSWER: 4. SHORT RESPONSE Suki is doing fashion design at 4-H Club. Her first project is to make a simple A-line skirt.
How much fabric will she need according to the design at the right?
SOLUTION: The area A of a trapezoid is A =
h(b 1 + b 2). She will need
· 2 or
square feet of fabric.
ANSWER: ALGEBRA Find x.
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5. Page 2
ANSWER: 11-2 Areas of Trapezoids, Rhombi, and Kites
ALGEBRA Find x.
5. SOLUTION: ANSWER: 8 cm
6. SOLUTION: ANSWER: 6.6 in.
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11-2ANSWER: Areas of Trapezoids, Rhombi, and Kites
6.6 in.
7. SOLUTION: ANSWER: 6.3 ft
CCSS STRUCTURE Find the area of each trapezoid, rhombus, or kite.
8. SOLUTION: h = 13, b 1 = 18 and b 2 = 24
ANSWER: 9. SOLUTION: h = 23, b 1 = 22 and b 2 = 37
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ANSWER: 11-2 Areas of Trapezoids, Rhombi, and Kites
9. SOLUTION: h = 23, b 1 = 22 and b 2 = 37
ANSWER: 10. SOLUTION: d 1 = 22 and d 2 = 24 ANSWER: 11. SOLUTION: d 1 = 16 in. and d 2 = 17 in. eSolutions Manual - Powered by Cognero
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ANSWER: 11-2 Areas of Trapezoids, Rhombi, and Kites
11. SOLUTION: d 1 = 16 in. and d 2 = 17 in. ANSWER: 12. SOLUTION: d 1 = 7 cm. and d 2 = 6 + 9 = 15 cm
ANSWER: 13. SOLUTION: d 1 = 11 ft. and d 2 = 25 ft eSolutions Manual - Powered by Cognero
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ANSWER: 11-2 Areas of Trapezoids, Rhombi, and Kites
13. SOLUTION: d 1 = 11 ft. and d 2 = 25 ft ANSWER: MICROSCOPES Find the area of the identified portion of each magnified image. Assume that the
identified portion is either a trapezoid, rhombus, or kite. Measures are provided in microns.
14. human skin
Refer to the image on Page 777.
SOLUTION: ANSWER: 26 square microns
15. heartleaf plant
Refer to the image on Page 777.
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ANSWER: 11-226
Areas
of microns
Trapezoids, Rhombi, and Kites
square
15. heartleaf plant
Refer to the image on Page 777.
SOLUTION: ANSWER: 24.5 square microns
16. eye of a fly
Refer to the image on Page 777.
SOLUTION: The figure consists of two trapezoids. ANSWER: 9.9 square microns
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17. JOBS
Jimmy
worksbyon
his neighbors’ yards
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after school to earn extra money to buy a car. He is going to plant grass
seed in Mr. Troyer’s yard. What is the area of the yard?
11-2ANSWER: Areas of Trapezoids, Rhombi, and Kites
9.9 square microns
17. JOBS Jimmy works on his neighbors’ yards after school to earn extra money to buy a car. He is going to plant grass
seed in Mr. Troyer’s yard. What is the area of the yard?
SOLUTION: ANSWER: ALGEBRA Find each missing length.
18. One diagonal of a kite is twice as long as the other diagonal. If the area of the kite is 240 square inches, what are the
lengths of the diagonals?
SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d 1 and d 2. Therefore, the diagonals are of length 15.5 in. and 31.0 in.
ANSWER: 15.5 in., 31.0 in.
19. The area of a rhombus is 168 square centimeters. If one diagonal is three times as long as the other, what are the
lengths of the diagonals?
SOLUTION: The area A of a rhombus is one half the product of the lengths of its diagonals, d 1 and d 2. eSolutions Manual - Powered by Cognero
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Therefore, the diagonals are of length 15.5 in. and 31.0 in.
ANSWER: 11-215.5
Areas
Trapezoids,
Rhombi, and Kites
in., of
31.0
in.
19. The area of a rhombus is 168 square centimeters. If one diagonal is three times as long as the other, what are the
lengths of the diagonals?
SOLUTION: The area A of a rhombus is one half the product of the lengths of its diagonals, d 1 and d 2. Therefore, the diagonals are of length 10.6 cm. and 31.7 cm.
ANSWER: 10.6 cm, 31.7 cm
20. A trapezoid has base lengths of 12 and 14 feet with an area of 322 square feet. What is the height of the trapezoid?
SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b 1 and b 2. .
ANSWER: 24.8 ft
21. A trapezoid has a height of 8 meters, a base length of 12 meters, and an area of 64 square meters. What is the
length of the other base?
SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b 1 and b 2. eSolutions Manual - Powered by Cognero
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.
11-2ANSWER: Areas of Trapezoids, Rhombi, and Kites
24.8 ft
21. A trapezoid has a height of 8 meters, a base length of 12 meters, and an area of 64 square meters. What is the
length of the other base?
SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b 1 and b 2. ANSWER: 4m
22. HONORS Estella has been asked to join an honor society at school. Before the first meeting, new members are
asked to sand and stain the front side of a piece of wood in the shape of an isosceles trapezoid. What is the surface
area that Allison will need to sand and stain?
SOLUTION: The required area is the difference between the larger trapezoid and the smaller trapezoid.
The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b 1 and b 2. ANSWER: For Manual
each figure,
a justification
eSolutions
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by Cognero
showing that
.
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ANSWER: 11-2 Areas of Trapezoids, Rhombi, and Kites
For each figure, provide a justification showing that
.
23. SOLUTION: Set the area of the kite equal to the sum of the areas of the two triangles with bases d 1 and d 2.
ANSWER: and the area of The area of
the area of . Therefore, the area of
. The area of kite FGHJ is equal to the area of
. After simplification, the area of kite FGHJ is equal to
, and
the area of
.
24. SOLUTION: Set the area of the kite equal to the sum of the areas of the two triangles with bases d 1 and d 2.
ZWX = ZYX by SSS
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the area of . The area of kite FGHJ is equal to the area of
. After simplification,
11-2 Areas of Trapezoids, Rhombi,
and Kites the area of kite FGHJ is equal to
the area of
.
24. SOLUTION: Set the area of the kite equal to the sum of the areas of the two triangles with bases d 1 and d 2.
ZWX = ZYX by SSS
ANSWER: The area of
the area of
and the area of
. Therefore, the area of
. The area of rhombus WXYZ is equal to the area of
. After simplification, the area of rhombus WXYZ is equal to
, and
the area of
.
25. CRAFTS Ashanti is in a kite competition. The yellow, red, orange, green, and blue pieces of her kite design shown
are all congruent rhombi.
a. How much fabric of each color does she need to buy?
b. Competition rules require that the total area of each kite be no greater than 200 square inches. Does Ashanti’s
kite meet this requirement? Explain.
SOLUTION: The area of the yellow rhombus is
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2
or 24 in . Since the yellow, red , orange, green, and blue pieces area all
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2
congruent rhombi, each have an area of 24 in . 2
the area of
. The area of rhombus WXYZ is equal to the area of
the area of
11-2 Areas of Trapezoids, Rhombi,
and Kites the area of rhombus WXYZ is equal to
. After simplification,
.
25. CRAFTS Ashanti is in a kite competition. The yellow, red, orange, green, and blue pieces of her kite design shown
are all congruent rhombi.
a. How much fabric of each color does she need to buy?
b. Competition rules require that the total area of each kite be no greater than 200 square inches. Does Ashanti’s
kite meet this requirement? Explain.
SOLUTION: 2
or 24 in . Since the yellow, red , orange, green, and blue pieces area all
The area of the yellow rhombus is
2
congruent rhombi, each have an area of 24 in . The area of the purple kite-shaped piece is
2
or 20 in .
2
2
The total area of the entire kite is 24(5) + 20 or 140 in , which is less than the maximum 200 in allowed. Therefore,
her kite meets this requirement.
ANSWER: a.
each of yellow, red, orange, green, and blue;
b. Yes; her kite has an area of
of purple
2
, which is less than 200 in .
CCSS SENSE-MAKING Find the area of each quadrilateral with the given vertices.
26. A(–8, 6), B(–5, 8), C(–2, 6), and D(–5, 0)
SOLUTION: Graph the quadrilateral.
The quadrilateral is a kite. The area A of a kite is one half the product of the lengths of its diagonals, d 1 and d 2. The
lengths of the diagonals are 6 units and 8 units. Therefore, the area of the kite is
ANSWER: 24 sq. units
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27. W(3, 0), X(0, 3), Y(–3, 0), and Z(0, –3)
SOLUTION: Page 14
lengths of the diagonals are 6 units and 8 units. Therefore, the area of the kite is
ANSWER: 11-2 Areas of Trapezoids, Rhombi, and Kites
24 sq. units
27. W(3, 0), X(0, 3), Y(–3, 0), and Z(0, –3)
SOLUTION: Graph the quadrilateral.
The quadrilateral is a rhombus. The area A of a rhombus is one half the product of the lengths of its diagonals, d 1
and d 2. The lengths of the diagonals are 6 units each. Therefore, the area of the kite is
ANSWER: 18 sq. units
28. METALS When magnified in very powerful microscopes, some metals are composed of grains that have various
polygonal shapes.
a. What is the area of figure 1 if the grain has a height of 4 microns and bases with lengths of 5 and 6 microns?
b. If figure 2 has perpendicular diagonal lengths of 3.8 microns and 4.9 microns, what is the area of the grain?
SOLUTION: a. Figure 1 is a trapezoid. The area of figure 1 is 22 square microns.
b. Figure 2 is a rhombus. The area of figure 2 is about 9.3 square microns.
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ANSWER: a. 22 square microns
b. 9.3 square microns
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The lengths of the diagonals are 6 units each. Therefore, the area of the kite is
11-2ANSWER: Areas of Trapezoids, Rhombi, and Kites
18 sq. units
28. METALS When magnified in very powerful microscopes, some metals are composed of grains that have various
polygonal shapes.
a. What is the area of figure 1 if the grain has a height of 4 microns and bases with lengths of 5 and 6 microns?
b. If figure 2 has perpendicular diagonal lengths of 3.8 microns and 4.9 microns, what is the area of the grain?
SOLUTION: a. Figure 1 is a trapezoid. The area of figure 1 is 22 square microns.
b. Figure 2 is a rhombus. The area of figure 2 is about 9.3 square microns.
ANSWER: a. 22 square microns
b. 9.3 square microns
29. PROOF The figure at the right is a trapezoid that consists of two congruent right triangles and an isosceles triangle.
In 1876, James A. Garfield, the 20th president of the United States, discovered a proof of the Pythagorean Theorem
using this diagram. Prove that .
SOLUTION: Following is an algebraic proof.
The trapezoid is on its side, so the bases are x and y and the height is x + y. Set the area of the trapezoid equal to the
sum of the areas of the triangles and simplify.
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The area of figure 2 is about 9.3 square microns.
ANSWER: 22 square
microns Rhombi, and Kites
11-2a.Areas
of Trapezoids,
b. 9.3 square microns
29. PROOF The figure at the right is a trapezoid that consists of two congruent right triangles and an isosceles triangle.
In 1876, James A. Garfield, the 20th president of the United States, discovered a proof of the Pythagorean Theorem
using this diagram. Prove that .
SOLUTION: Following is an algebraic proof.
The trapezoid is on its side, so the bases are x and y and the height is x + y. Set the area of the trapezoid equal to the
sum of the areas of the triangles and simplify.
ANSWER: The area of a trapezoid is
So, The area of
.
. The area of
. Set the area of the trapezoid equal to the combined areas of the triangles
to get
When simplified,
. Multiply by 2 on each side:
.
.
DIMENSIONAL ANALYSIS Find the perimeter and area of each figure in feet. Round to the nearest
tenth, if necessary.
30. SOLUTION: Both diagonals are perpendicular bisectors, so the figure is a rhombus and all four triangles are congruent. All of the
sides are 12 feet, so the perimeter is 48 feet.
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Use trigonometry to find the lengths of the diagonals.
. Set the area of the trapezoid equal to the combined areas of the triangles
to get
11-2 Areas of Trapezoids, Rhombi, and Kites
When simplified,
.
. Multiply by 2 on each side:
.
DIMENSIONAL ANALYSIS Find the perimeter and area of each figure in feet. Round to the nearest
tenth, if necessary.
30. SOLUTION: Both diagonals are perpendicular bisectors, so the figure is a rhombus and all four triangles are congruent. All of the
sides are 12 feet, so the perimeter is 48 feet.
Use trigonometry to find the lengths of the diagonals.
Now find the area.
ANSWER: 2
48 ft; 129.4 ft
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ANSWER: 2
11-2 Areas of Trapezoids,
Rhombi, and Kites
48 ft; 129.4 ft
31. SOLUTION: Use the 30-60-90 triangle to find the dimensions of the isosceles trapezoid. The base of the triangle is 0.5(12 – 8) =
2.
Don't forget to use dimensional analysis to convert the units to feet.
ANSWER: 2
2.3 ft; 0.24 ft
32. SOLUTION: The figure is a kite because one of the diagonals is a perpendicular bisector. Find the perimeter. Convert the units to
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32. 11-2 Areas of Trapezoids, Rhombi, and Kites
SOLUTION: The figure is a kite because one of the diagonals is a perpendicular bisector. Find the perimeter. Convert the units to
feet.
Use the 45-45-90 triangle to find the lengths of the congruent parts of the diagonals.
d1 = 3 + 3 = 6
Use the Pythagorean theorem to find the other piece of d 2.
Now find the area of the kite.
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11-2 Areas of Trapezoids, Rhombi, and Kites
Now find the area of the kite.
ANSWER: 67.6 ft; 267.8 ft
2
33. MULTIPLE REPRESENTATIONS In this problem, you will investigate perimeters of kites.
a. GEOMETRIC Draw a kite like the one shown if x = 2.
b. GEOMETRIC Repeat the process in part a for three x-values between 2 and 10 and for an x-value of 10.
c. TABULAR Measure and record in a table the perimeter of each kite, along with the x-value.
d. GRAPHICAL Graph the perimeter versus the x-value using the data from your table.
e. ANALYTICAL Make a conjecture about the value of x that will minimize the perimeter of the kite. What is the
significance of this value? SOLUTION: a. When x = 2, the long diagonal will be in pieces of length 2 and 10.
b. When x = 4, the long diagonal will be in pieces of length 4 and 8.
When x = 6, the long diagonal will be in pieces of length 6 and 6.
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11-2 Areas of Trapezoids, Rhombi, and Kites
When x = 6, the long diagonal will be in pieces of length 6 and 6.
When x = 8, the long diagonal will be in pieces of length 8 and 4.
When x = 10, the long diagonal will be in pieces of length 10 and 2.
c. Use the Pythagorean Theorem to first find the lengths of the sides of the kite and then find its perimeter.
For x = 2: Two sides of the kite are the length of the hypotenuse of right triangle with legs of 2 cm each and the
other two sides are the length of the hypotenuse of a right triangle with legs of 2 cm and 10 cm. Thus, the length of
two legs =
kite is
and the length of the other two sides =
cm.
. So, the perimeter of the
For x = 4: Two sides of the kite are the length of the hypotenuse of right triangle with legs of 2 cm and 4 cm and the
other two sides are the length of the hypotenuse of a right triangle with legs of 2 cm and 8 cm. Thus, the length of
two legs =
kite is
and the length of the other two sides =
cm.
. So, the perimeter of the
For x = 6: Two sides of the kite are the length of the hypotenuse of right triangle with legs of 2 cm and 6 cm and the
other two sides are the length of the hypotenuse of a right triangle with legs of 2 cm and 6 cm. Thus, the length of
two legs =
kite is
and the length of the other two sides =
cm.
. So, the perimeter of the
For x = 8: Two sides of the kite are the length of the hypotenuse of right triangle with legs of 2 cm and 8 cm and the
other two sides are the length of the hypotenuse of a right triangle with legs of 2 cm and 4 cm. Thus, the length of
two legs =
kite is
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and the length of the other two sides =
cm.
. So, the perimeter of the
Page 22
For x = 10: Two sides of the kite are the length of the hypotenuse of right triangle with legs of 2 cm and 10 cm and
the other two sides are the length of the hypotenuse of a right triangle with legs of 2 cm each. Thus, the length of
For x = 8: Two sides of the kite are the length of the hypotenuse of right triangle with legs of 2 cm and 8 cm and the
other two sides are the length of the hypotenuse of a right triangle with legs of 2 cm and 4 cm. Thus, the length of
11-2two
Areas
Kites of the other two sides =
legsof= Trapezoids, Rhombi,
and and
the length
kite is
cm.
. So, the perimeter of the
For x = 10: Two sides of the kite are the length of the hypotenuse of right triangle with legs of 2 cm and 10 cm and
the other two sides are the length of the hypotenuse of a right triangle with legs of 2 cm each. Thus, the length of
two legs =
kite is
and the length of the other two sides =
cm.
. So, the perimeter of the
x
2 cm
4 cm
6 cm
8 cm
10 cm
P
26.1
25.4
25.3
25.4
26.1
d.
e . Sample answer: Based on the graph, the perimeter will be minimized when x = 6. This value is significant because
when x = 6, the figure is a rhombus.
ANSWER: a.
b.
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11-2 Areas of Trapezoids, Rhombi, and Kites
b.
c.
x
2 cm
4 cm
6 cm
8 cm
10 cm
P
26.1
25.4
25.3
25.4
26.1
d.
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e . Sample answer: Based on the graph, the perimeter will be minimized when x = 6. This value is significant because
6 cm
25.3
8 cm
25.4
26.1 and Kites
11-2 Areas10ofcm
Trapezoids, Rhombi,
d.
e . Sample answer: Based on the graph, the perimeter will be minimized when x = 6. This value is significant because
when x = 6, the figure is a rhombus.
34. CCSS CRITIQUE Antonio and Madeline want to draw a trapezoid that has a height of 4 units and an area of 18
square units. Antonio says that only one trapezoid will meet the criteria. Madeline disagrees and thinks that she can
draw several different trapezoids with a height of 4 units and an area of 18 square units. Is either of them correct?
Explain your reasoning.
SOLUTION: There is more than one trapezoid with a height of 4 units and an area of 18 square units. The sum of the bases of the
trapezoid has to be 9, so one possibility is a trapezoid with bases of 4 and 5 units and a height of 4 units. Another is a
trapezoid with bases of 3 and 6 units and a height of 4 units. Therefore, Madeline is correct.
ANSWER: Madeline; sample answer: There is more than one trapezoid with a height of 4 units and an area of 18 square units.
The sum of the bases of the trapezoid has to be 9, so one possibility is a trapezoid with bases of 4 and 5 units and a
height of 4 units. Another is a trapezoid with bases of 3 and 6 units and a height of 4 units.
35. CHALLENGE Find x in parallelogram ABCD.
SOLUTION: Let the two segments of the side
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be a and 15 – a.
Page 25
ANSWER: Madeline; sample answer: There is more than one trapezoid with a height of 4 units and an area of 18 square units.
sumofofTrapezoids,
the bases of the
trapezoid
to be 9, so one possibility is a trapezoid with bases of 4 and 5 units and a
11-2The
Areas
Rhombi,
andhas
Kites
height of 4 units. Another is a trapezoid with bases of 3 and 6 units and a height of 4 units.
35. CHALLENGE Find x in parallelogram ABCD.
SOLUTION: Let the two segments of the side
be a and 15 – a.
We have two variables, x and a. They are a part of two right triangles, so we can use the Pythagorean Theorem for
each triangle and then combine the equations.
We now have x2 on the left side of each equation. Use substitution.
Use the value of a to find the value of x.
ANSWER: 7.2
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OPEN
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36. Draw a kite and a rhombus with an area of 6 square inches. Label and justify your drawings.
SOLUTION: Page 26
11-2ANSWER: Areas of Trapezoids, Rhombi, and Kites
7.2
36. OPEN ENDED Draw a kite and a rhombus with an area of 6 square inches. Label and justify your drawings.
SOLUTION: Sample answer: Since the area formula for both a rhombus and a kite is one half the product of the lengths of the
two diagonals, if the area is 6 square inches, the product of the two diagonals must be 12. I used 3 and 4 inches for
the diagonals of the rhombus and 2 and 6 inches for the diagonals of the kite.
Both diagonals must be perpendicular bisectors for the rhombus and one diagonal must be a perpendicular bisector
for the kite.
ANSWER: Sample answer: Since the area formula for both a rhombus and a kite is one half the product of the lengths of the
two diagonals, if the area is 6 square inches, the product of the two diagonals must be 12. I used 3 and 4 inches for
the diagonals of the rhombus and 2 and 6 inches for the diagonals of the kite.
37. REASONING If the areas of two rhombi are equal, are the perimeters sometimes, always, or never equal?
Explain.
SOLUTION: If the areas are equal, it means that the products of the diagonals are equal. The only time that the perimeters will be
equal is when the diagonals are also equal, or when the two rhombi are congruent. Therefore, the statement is
sometimes true.
For example,
When the diagonals are 6 and 8, the area is 24 and the perimeter is 26.
When the diagonals are 12 and 4, the area is 24 and the perimeter is 32.
ANSWER: Sometimes; sample answer: If the areas are equal, it means that the products of the diagonals are equal. The only
time that the perimeters will be equal is when the diagonals are also equal, or when the two rhombi are congruent.
38. WRITING IN MATH How can you use trigonometry to find the area of a figure?
SOLUTION: Trigonometry can be used with known angles and sides to find unknown sides. For example;
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Page 27
When the diagonals are 12 and 4, the area is 24 and the perimeter is 32.
ANSWER: sample answer:
If the areas
are equal, it means that the products of the diagonals are equal. The only
11-2Sometimes;
Areas of Trapezoids,
Rhombi,
and Kites
time that the perimeters will be equal is when the diagonals are also equal, or when the two rhombi are congruent.
38. WRITING IN MATH How can you use trigonometry to find the area of a figure?
SOLUTION: Trigonometry can be used with known angles and sides to find unknown sides. For example;
The height of the parallelogram can be found using the sine function:
And the area can be found as usual:
ANSWER: Sample answer: You can use trigonometry and known angle and side measures to find unknown triangular measures
that are required to calculate the area.
39. The lengths of the bases of an isosceles trapezoid are shown below.
If the perimeter is 74 meters, what is its area?
A
B C
D
SOLUTION: The lengths of the legs of an isosceles trapezoid are congruent. The perimeter is 74 and the sum of the lengths of the
bases is 54 m, so the sides are each 10 m.
We can draw dashed lines for the height to produce two congruent triangles
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The bases of the triangles are 8 because the middle figure is a rectangle and the entire base is 35.
Page 28
SOLUTION: The lengths of the legs of an isosceles trapezoid are congruent. The perimeter is 74 and the sum of the lengths of the
bases is 54 m, so the sides are each 10 m.
11-2 Areas of Trapezoids, Rhombi, and Kites
We can draw dashed lines for the height to produce two congruent triangles
The bases of the triangles are 8 because the middle figure is a rectangle and the entire base is 35.
Use the Pythagorean Theorem to find the height h.
Now find the area of the trapezoid.
ANSWER: A
40. SHORT RESPONSE One diagonal of a rhombus is three times as long as the other diagonal. If the area of the
rhombus is 54 square millimeters, what are the lengths of the diagonals?
SOLUTION: Therefore, the diagonals are of length 6 mm. and 18 mm.
ANSWER: 6 mm, 18 mm
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11-2ANSWER: Areas of Trapezoids, Rhombi, and Kites
A
40. SHORT RESPONSE One diagonal of a rhombus is three times as long as the other diagonal. If the area of the
rhombus is 54 square millimeters, what are the lengths of the diagonals?
SOLUTION: Therefore, the diagonals are of length 6 mm. and 18 mm.
ANSWER: 6 mm, 18 mm
41. ALGEBRA What is the effect on the graph of the equation
when the equation is changed to y = –2x?
F The graph is moved 1 unit down.
G The graph is moved 1 unit up.
H The graph is rotated 45° about the origin.
J The graph is rotated 90° about the origin.
SOLUTION: The product of the slopes of the two lines is –1. So, the lines are perpendicular to each other. When two lines are perpendicular, the first line is rotated 90° about the origin to get the second line. Therefore, the correct choice is J.
ANSWER: J
42. A regular hexagon is divided into 6 congruent triangles. If the perimeter of the hexagon is 48 centimeters, what is the
height of each triangle?
A 4 cm
B
cm
C
cm
D 8 cm
E
cm
SOLUTION: The perimeter of the hexagon is 48 cm, so the length of each side of the hexagon is 8 cm. The height of each triangle
is equal to the apothem of the hexagon. Each interior angle of a regular hexagon measures
So, each
diagonal will make an angle of 60° with the sides. An apothem bisects the side of the polygon.
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The triangle with the sides of one apothem, half the side of the polygon to which the apothem is drawn, and the line
joining the center and the vertex at endpoints of the side form a
triangle.
Therefore, the correct choice is J.
11-2ANSWER: Areas of Trapezoids, Rhombi, and Kites
J
42. A regular hexagon is divided into 6 congruent triangles. If the perimeter of the hexagon is 48 centimeters, what is the
height of each triangle?
A 4 cm
B
cm
C
cm
D 8 cm
E
cm
SOLUTION: The perimeter of the hexagon is 48 cm, so the length of each side of the hexagon is 8 cm. The height of each triangle
is equal to the apothem of the hexagon. Each interior angle of a regular hexagon measures
So, each
diagonal will make an angle of 60° with the sides. An apothem bisects the side of the polygon.
The triangle with the sides of one apothem, half the side of the polygon to which the apothem is drawn, and the line
joining the center and the vertex at endpoints of the side form a
triangle.
Let h be the height of the triangle. Use the tangent ratio to find h.
The height of each triangle is
.
ANSWER: B
COORDINATE GEOMETRY Find the area of each figure.
43. JKL with J(–4, 3), K(–9, –1), and L(–4, –4)
SOLUTION: Plot the points and draw the triangle.
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The height of each triangle is
.
11-2ANSWER: Areas of Trapezoids, Rhombi, and Kites
B
COORDINATE GEOMETRY Find the area of each figure.
43. JKL with J(–4, 3), K(–9, –1), and L(–4, –4)
SOLUTION: Plot the points and draw the triangle.
The length of base JL is 7 units. The corresponding height is from the line x = –9 to x = –4, so the height is 5 units. ANSWER: 44. RSTV with R(–5, 7), S(2, 7), T(0, 2), and V(–7, 2)
SOLUTION: Plot the points.
The parallelogram has base length of 7 units.
2
The height is from y = 2 to y = 7, so the height is 5 units. The area of the parallelogram is 7(5) or 35 units .
ANSWER: Meteorologists
45. WEATHER
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track severe storms using Doppler radar. A polar grid is used to measure distancesPage
as 32
the storms progress. If the center of the radar screen is the origin and each ring is 10 miles farther from the center,
what is the equation of the fourth ring?
Refer to the image on Page 780.
2
The height is from y = 2 to y = 7, so the height is 5 units. The area of the parallelogram is 7(5) or 35 units .
ANSWER: 11-2 Areas of Trapezoids, Rhombi, and Kites
45. WEATHER Meteorologists track severe storms using Doppler radar. A polar grid is used to measure distances as
the storms progress. If the center of the radar screen is the origin and each ring is 10 miles farther from the center,
what is the equation of the fourth ring?
Refer to the image on Page 780.
SOLUTION: 2
2
2
The standard form of the equation of a circle with center at (h, k) and radius r is (x – h) + (y – k) = r .
(h, k) = (0, 0) The fourth ring will be 40 miles away from the origin, so r = 40. 2
2
2
2
2
Therefore, the equation is (x – 0) + (y – 0) = 40 or x + y = 1600.
ANSWER: Find x and y.
46. SOLUTION: This is a 30°-60°-90° triangle.
Side opposite the 30° = a
Side opposite the 60° = Hypotenuse = 2a
For this triangle:
Side opposite the 30° = 8
Side opposite the 60° = Hypotenuse = 16
Therefore, x =
and y = 12
ANSWER: 47. SOLUTION: This is a 30°-60°-90° triangle.
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Side opposite the 30° = a
Side opposite the 60° = Page 33
Therefore, x =
and y = 12
ANSWER: 11-2 Areas of Trapezoids, Rhombi, and Kites
47. SOLUTION: This is a 30°-60°-90° triangle.
Side opposite the 30° = a
Side opposite the 60° = Hypotenuse = 2a
For this triangle:
Side opposite the 30° = 9
Side opposite the 60° = Hypotenuse = 18
Therefore, x = 9 and y =
ANSWER: Use the Venn diagram to determine whether each statement is always, sometimes, or never true.
48. A parallelogram is a square.
SOLUTION: When a figure is a parallelogram, it can be anything inside the yellow sphere of the Venn diagram. The figure is a
square only when it is inside the purple intersection in the middle of the diagram. Therefore, a figure that is a
parallelogram is sometimes also a square.
ANSWER: sometimes
49. A square is a rhombus.
SOLUTION: Page 34
When a figure is a square, it is inside the purple intersection in the middle of the diagram. This region is within the
region of a rhombus, so when a figure is a square, it is always a rhombus as well. eSolutions Manual - Powered by Cognero
ANSWER: 11-2sometimes
Areas of Trapezoids, Rhombi, and Kites
49. A square is a rhombus.
SOLUTION: When a figure is a square, it is inside the purple intersection in the middle of the diagram. This region is within the
region of a rhombus, so when a figure is a square, it is always a rhombus as well. ANSWER: always
50. A rectangle is a parallelogram.
SOLUTION: When a figure is a rectangle, it can be anything inside the blue sphere of the Venn diagram. This region is within the
region of a parallelogram, so any figure that is a rectangle is always also a parallelogram. ANSWER: always
51. A rhombus is a rectangle but not a square.
SOLUTION: When a figure is a rhombus, it can be anything inside the red sphere of the Venn diagram. The figure is a rectangle
when it is inside the blue sphere. The intersection of these two spheres indicate when a figure is both a rhombus and
a rectangle. However, everything within this region is also a square, so a figure can never be both a rhombus and a
rectangle while not also being a square.
ANSWER: never
52. A rhombus is a square.
SOLUTION: When a figure is a rhombus, it can be anything inside the red sphere of the Venn diagram. When a figure is within
the purple region, it is a square. However, a figure within the red region but not within the purple region is a rhombus
that is not a square.
Therefore, a figure can sometimes be both a rhombus and a square.
ANSWER: sometimes
Find the circumference and area of each figure. Round to the nearest tenth.
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53. SOLUTION: Page 35
11-2ANSWER: Areas of Trapezoids, Rhombi, and Kites
sometimes
Find the circumference and area of each figure. Round to the nearest tenth.
53. SOLUTION: ANSWER: 54. SOLUTION: ANSWER: eSolutions Manual - Powered by Cognero
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ANSWER: 11-2 Areas of Trapezoids, Rhombi, and Kites
55. SOLUTION: ANSWER: eSolutions Manual - Powered by Cognero
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